The Conjecture
Consider k runners on a circular track of unit length. At t = 0, all runners are at the same position and start to run; the runners' speeds are pairwise distinct. A runner is said to be lonely at time t if he is at distance of at least 1/k from every other runner at time t. The lonely runner conjecture states that each runner is lonely at some time.
A convenient reformulation of the problem is to assume that the runners have integer speeds, not all divisible by the same prime; the runner to be lonely has zero speed. The conjecture then states that for any set D of k-1 positive integers with gcd 1,
where ||x|| denotes the distance of real number x to the nearest integer.
Read more about this topic: Lonely Runner Conjecture
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